- Titel
- Analysis and Numerics of Stochastic Gradient Flows
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-807454
- Datum der Einreichung
- 25.03.2022
- Datum der Verteidigung
- 20.09.2022
- Abstract (EN)
- In this thesis we study three stochastic partial differential equations (SPDE) that arise as stochastic gradient flows via the fluctuation-dissipation principle. For the first equation we establish a finer regularity statement based on a generalized Taylor expansion which is inspired by the theory of rough paths. The second equation is the thin-film equation with thermal noise which is a singular SPDE. In order to circumvent the issue of dealing with possible renormalization, we discretize the gradient flow structure of the deterministic thin-film equation. Choosing a specific discretization of the metric tensor, we resdiscover a well-known discretization of the thin-film equation introduced by Grün and Rumpf that satisfies a discrete entropy estimate. By proving a stochastic entropy estimate in this discrete setting, we obtain positivity of the scheme in the case of no-slip boundary conditions. Moreover, we analyze the associated rate functional and perform numerical experiments which suggest that the scheme converges. The third equation is the massive $\varphi^4_2$-model on the torus which is also a singular SPDE. In the spirit of Bakry and Émery, we obtain a gradient bound on the Markov semigroup. The proof relies on an $L^2$-estimate for the linearization of the equation. Due to the required renormalization, we use a stopping time argument in order to ensure stochastic integrability of the random constant in the estimate. A postprocessing of this estimate yields an even sharper gradient bound. As a corollary, for large enough mass, we establish a local spectral gap inequality which by ergodicity yields a spectral gap inequality for the $\varphi^4_2$- measure.
- Freie Schlagwörter (DE)
- singular stochastic partial differential equations, stochastic gradient flows, gradient bounds, regularity theory
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- angenommene Version / Postprint / Autorenversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-807454
- Veröffentlichungsdatum Qucosa
- 22.09.2022
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
CC BY-NC-ND 4.0